Problem: Amir throws a stone off of a bridge into a river. The stone's height (in meters above the water) $t$ seconds after Amir throws it is modeled by $h(t)=-5t^2+20t+160$ Amir wants to know when the stone will reach its highest point. 1) Rewrite the function in a different form (factored or vertex) where the answer appears as a number in the equation. $h(t)=$ 2) How many seconds after being thrown did the stone reach its highest point?
Choosing a form The stone's highest point relates to the maximum of the function. Which form reveals this feature? Here's a summary of what each form reveals along with examples. Note that these are all equivalent forms of the same function, but not the function modeling the height of the stone. Form Example Feature revealed Standard $f(x)=2x^2-12x+{10}$ $y$ -intercept is ${10}$ Factored $f(x)=2(x-C{1})(x-C{5})$ Zeros are $x=C1$ and $x=C5$ Vertex $f(x)=2(x-{3})^2{-8}$ Vertex is $(3,{-8})$ Rewrite in vertex form The vertex of the function tells us the value of $t$ where the function reaches its maximum height, so let's rewrite $h(t)$ in vertex form by completing the square. The number that will help us complete the square is $\left(\dfrac{{-4}}{2}\right)^2={4}$ : $\begin{aligned} h(t)&=-5t^2+20t+160 \\\\ &=-5(t^2-4t)+160&&\text{Factor } -5 \text{ from first two terms}. \\\\ &=-5(t^2{-4}t+{4})+160{+20}&&\text{Complete the square}. \\\\ &=-5(t-2)^2+180&&\text{Factor and simplify}. \end{aligned}$ [How do we know what to add to complete the square?] When did the stone reach its highest point? The vertex form of the function reveals its vertex, and we know this point is a maximum for $h(t)$ since the leading coefficient $-5$ is negative. In general, for any quadratic function written in vertex form $f(x)=a(x-{h})^2+{k}$, we can conclude that the vertex is the point $({h},{k})$. So for $h(t)=-5(t-{2})^2+{180}$, the vertex is $({2},{180})$, and we know the stone reached its highest point at $t={2}$ seconds. Answers 1) The vertex form of the function reveals when the stone reached its highest point: $h(t)=-5\left(t-2\right)^2+180$ 2) The stone reached its highest point $2$ seconds after being thrown.